Electricity on a spacecraft can be produced by a type of nuclear generator - Scottish Highers Maths - Question 9 - 2019

To find how long it takes for the electrical power to reduce by 15%, we need to determine the power at which the generator runs after this reduction:

$P = 120 - 0.15(120) = 120 - 18 = 102 ext{ watts}.$

Now we set up the equation using the model:

$P_t = 120 e^{-0.007t}$

Substituting the known value:

$102 = 120 e^{-0.007t}.$

Dividing both sides by 120 gives us:

e^{-0.007t} = rac{102}{120} = 0.85.

Next, we take the natural logarithm of both sides:

$-0.007t = ext{ln}(0.85).$

Solving for $t$ results in:

t = rac{ ext{ln}(0.85)}{-0.007}.

Using a calculator, we find:

t ext{ (approximately)} = rac{-0.1625}{-0.007} ext{ which is about } 23.21 ext{ years.}

Hence, it takes approximately 23.2 years for the electrical power to reduce by 15%.